ENGINEERING DRAWING | Conic Sections and Interpenetration of Solids

harry w

May 3

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Consider a right circular cone, i.e. a cone whose base is a circle and whose apex is above the centre of the base (Fig. 12.1). The true face of a section through the apex of the cone will be a triangle.

The true face of a section drawn parallel to the base will be a circle.

The true face of any other section which passes through two opposite generators will be an ellipse.

The true face of a section drawn parallel to the generator will be a parabola.

If a plane cuts the cone through the generator and the base on the same side of the cone axis, then a view on the true face of the section will be a hyperbola. The special case of a section at right-angles to the base gives a rectangular hyperbola.

To Draw an Ellipse from Part of a Cone

Figure 12.2 shows the method of drawing the ellipse, which is a true view on the surface marked AB of the frustum of the given cone.

Draw a centre line parallel to line AB as part of an auxiliary view.
Project points A and B onto this line and onto the centre lines of the plan and end elevation.
Take any horizontal section XX between A and B and draw a circle in the plan view of diameter D
Project the line of section plane XX onto the end elevation.
Project the point of intersection of line AB and plane XX onto the plan view.
Mark the chord-width W on the plan, in the auxiliary view and the end elevation. These points in the auxiliary view form part of the ellipse.
Repeat with further horizontal sections between A and B, to complete the views as shown.

To Draw a Parabola from Part of a Cone

Figure 12.3 shows the method of drawing the parabola, which is a true view on the line AB drawn parallel to the sloping side of the cone.

Draw a centre line parallel to line AB as part of an auxiliary view.
Project point B to the circumference of the base in the plan view, to give the points B₁ and B₂. Mark chord-width B₁B₂ in the auxiliary view and in the end elevation.
Project point A onto the other three views.
Take any horizontal section XX between A and B and draw a circle in the plan view of diameter D.
Project the line of section plane XX onto the end elevation.
Project the point of intersection of line AB and plane XX to the plane view.
Mark the chord-width W on the plan, in the end elevation and the auxiliary view. These points in the auxiliary view form part of the parabola.
Repeat with further horizontal sections between A and B, to complete the three views.

To Draw a Rectangular Hyperbola from Part of a Cone

Figure 12.4 shows the method of drawing the hyperbola, which is a true view on the line AB drawn parallel to the vertical centre line of the cone.

Project point B to the circumference of the base in the plan view, to give the points B1 and B2.
Mark points B1 and B2 in the end elevation.
Project point A onto the end elevation. Point A lies on the centre line in the plan view.
Take any horizontal section XX between A and B and draw a circle of diameter D in the plan view.
Project the line of section XX onto the end elevation.
Mark the chord-width W in the plan, on the end elevation. These points in the end elevation form part of the hyperbola.
Repeat with further horizontal sections between A and B, to complete the hyperbola.

The ellipse, parabola, and hyperbola are also the loci of points which move in fixed ratios from a line (the directrix) and a point (the focus). The ratio is known as the eccentricity.

The eccentricity for the ellipse is less than one.
The eccentricity for the parabola is one.
The eccentricity for the hyperbola is greater than one.

Figure 12.5 shows an ellipse of eccentricity 3/5, a parabola of eccentricity 1, and a hyperbola of eccentricity 5/3. The distances from the focus are all radial, and the distances from the directrix are perpendicular, as shown by the illustration.

To assist in the construction of the ellipse in Fig. 12.5, the following method may be used to ensure that the two dimensions from the focus and directrix are in the same ratio. Draw triangle PA1 so that side A1 and side P1 are in the ratio of 3 units to 5 units. Extend both sides as shown. From any points B, C, D, etc., draw vertical lines to meet the horizontal at 2, 3, 4,
etc.; by similar triangles, vertical lines and their corresponding horizontal lines will be in the same ratio. A similar construction for the hyperbola is shown in Fig. 12.6.

Commence the construction for the ellipse by drawing a line parallel to the directrix at a perpendicular distance of P3 (Fig. 12.6 (a)). Draw radius C3 from point F1 to

intersect this line. The point of intersection lies on the ellipse. Similarly, for the hyperbola (Fig. 12.6 (b)) draw a line parallel to the directrix at a perpendicular distance of Q2. Draw radius S2, and the hyperbola passes through the point of intersection. No scale is required for the parabola, as the perpendicular distances and the radii are the same magnitude.

Repeat the procedure in each case to obtain the required curves.

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Interpenetration

Many objects are formed by a collection of geometrical shapes such as cubes, cones, spheres, cylinders, prisms, pyramids, etc., and where any two of these shapes meet, some sort of curve of intersection or interpenetration results. It is necessary to be able to draw these curves to complete drawings in orthographic projection or to draw patterns and developments.

The following drawings show some of the most commonly found examples of interpenetration.
Basically, most curves are constructed by taking sections through the intersecting shapes, and, to keep construction lines to a minimum and hence avoid confusion, only one or two sections have been taken in arbitrary positions to show the principle involved; further similar parallel sections are then required to establish the line of the curve in its complete form. Where centre lines are offset, hidden curves will not be the same as curves directly facing the draughtsman, but the draughting principle of taking sections in the manner indicated on either side of the centre lines of the shapes involved will certainly be the same.

If two cylinders, or a cone and a cylinder, or two cones intersect each other at any angle, and the curved surfaces of both solids enclose the same sphere, then the outline of the intersection in each case will be an ellipse. In the illustrations given in Fig. 12.7 the centre lines of the two solids intersect at point O, and a true view along the line AB will produce an ellipse.

When cylinders of equal diameter intersect as shown in Fig. 12.8 the line at the intersection is straight and at 45°

Figure 12.9 shows a branch cylinder square with the axis of the vertical cylinder but reduced in size. A section through any cylinder parallel with the axis produces a rectangle, in this case of width Y in the branch and width X in the vertical cylinder. Note that interpenetration occurs at points marked 3, and these points lie on a curve. The projection of the branch cylinder along the horizontal centre line gives the points marked 1, and along the vertical centre line gives the points marked 2.

Figure 12.10 shows a cylinder with a branch on the same vertical centre line but inclined at an angle. Instead of an end elevation, the position of section AA is shown on a part auxiliary view of the branch. The construction is otherwise the same as that for Fig. 12.9.

In Fig. 12.11 the branch is offset, but the construction is similar to that shown in Fig. 12.10.

Figure 12.12 shows the branch offset but square with the vertical axis.

Figure 12.13 shows a cone passing through a cylinder. A horizontal section AA through the cone will give a circle of ØP, and through the cylinder will give a rectangle of width X. The points of intersection of the circle and part of the rectangle in the plan view are projected up to the section plane in the front elevation.

The plotting of more points from more sections will give the interpenetration curves shown in the front elevation and the plan.

Figure 12.14 shows a cylinder passing through a cone. The construction shown is the same as for Fig. 12.13 in principle.

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Figure 12.15 shows a cone and a square prism where interpenetration starts along the horizontal section BB at point 1 on the smallest diameter circle to touch the prism. Section AA is an arbitrary section where the projected diameter of the cone ØX cuts the prism in the plan view at the points marked 2. These points are then projected back to the section plane in the front elevation and lie on the curve required. The circle at section CC is the largest circle which will touch the prism across the diagonals in the plan view. Having drawn the circle in the plan view, it is projected up to the sides of the cone in the front elevation, and points 3 at the corners of the prism are the lowest points of contact.

A casting with a rectangular base and a circular-section shaft is given in Fig. 12.16. The machining of the radius R1 in conjunction with the milling of the flat surfaces produces the curve shown in the front elevation. Point 1 is shown projected from the end elevation. Section AA produces a circle of ØX in the plan view and cuts the face of the casting at points marked 2, which are transferred back to the section plane. Similarly, section BB gives ØY and points marked 3. Sections can be taken until the circle in the plane view increases in size to R2; at this point, the interpenetration curve joins a horizontal line to the corner of the casting in the front elevation.

In Fig. 12.17 a circular bar of diameter D has been turned about the centre line CC and machined with a radius shown as RAD A. The resulting interpenetration curve is obtained by taking sections similar to section XX. At this section plane, a circle of radius B is projected
in the front elevation and cuts the circumference of the bar at points E and F. The projection of point F along the section plane XX is one point on the curve. By taking a succession of sections, and repeating the process described, the curve can be plotted.

Note that, in all these types of problem, it rarely helps to take dozens of sections and then draw all the circles before plotting the points, as the only result is possible confusion. It is recommended that one section be taken at a time, the first roughly near the centre of
any curve, and others sufficiently far apart for clarity but near enough to maintain accuracy. More sections are generally required where curves suddenly change direction.

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